| L(s) = 1 | − 1.25e9·2-s + 1.00e18·4-s + 4.05e20·5-s + 2.66e24·7-s − 5.43e26·8-s − 5.09e29·10-s − 7.79e30·11-s + 1.35e32·13-s − 3.35e33·14-s + 1.02e35·16-s + 2.91e36·17-s − 6.13e36·19-s + 4.08e38·20-s + 9.81e39·22-s + 2.47e39·23-s − 9.34e39·25-s − 1.71e41·26-s + 2.68e42·28-s + 4.74e42·29-s + 1.04e43·31-s + 1.83e44·32-s − 3.67e45·34-s + 1.07e45·35-s − 7.35e45·37-s + 7.71e45·38-s − 2.20e47·40-s + 6.31e47·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.74·4-s + 0.972·5-s + 0.312·7-s − 1.24·8-s − 1.61·10-s − 1.48·11-s + 0.187·13-s − 0.518·14-s + 0.309·16-s + 1.46·17-s − 0.115·19-s + 1.70·20-s + 2.45·22-s + 0.167·23-s − 0.0538·25-s − 0.310·26-s + 0.546·28-s + 0.343·29-s + 0.106·31-s + 0.728·32-s − 2.43·34-s + 0.304·35-s − 0.402·37-s + 0.192·38-s − 1.20·40-s + 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(60-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+59/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(30)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{61}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.25e9T + 5.76e17T^{2} \) |
| 5 | \( 1 - 4.05e20T + 1.73e41T^{2} \) |
| 7 | \( 1 - 2.66e24T + 7.25e49T^{2} \) |
| 11 | \( 1 + 7.79e30T + 2.76e61T^{2} \) |
| 13 | \( 1 - 1.35e32T + 5.28e65T^{2} \) |
| 17 | \( 1 - 2.91e36T + 3.94e72T^{2} \) |
| 19 | \( 1 + 6.13e36T + 2.79e75T^{2} \) |
| 23 | \( 1 - 2.47e39T + 2.19e80T^{2} \) |
| 29 | \( 1 - 4.74e42T + 1.91e86T^{2} \) |
| 31 | \( 1 - 1.04e43T + 9.78e87T^{2} \) |
| 37 | \( 1 + 7.35e45T + 3.34e92T^{2} \) |
| 41 | \( 1 - 6.31e47T + 1.42e95T^{2} \) |
| 43 | \( 1 + 5.26e47T + 2.36e96T^{2} \) |
| 47 | \( 1 + 3.33e49T + 4.50e98T^{2} \) |
| 53 | \( 1 + 7.06e50T + 5.39e101T^{2} \) |
| 59 | \( 1 + 2.46e52T + 3.02e104T^{2} \) |
| 61 | \( 1 - 5.32e52T + 2.16e105T^{2} \) |
| 67 | \( 1 + 2.99e53T + 5.47e107T^{2} \) |
| 71 | \( 1 - 4.94e54T + 1.67e109T^{2} \) |
| 73 | \( 1 - 7.41e54T + 8.63e109T^{2} \) |
| 79 | \( 1 + 6.76e55T + 9.12e111T^{2} \) |
| 83 | \( 1 + 4.29e56T + 1.68e113T^{2} \) |
| 89 | \( 1 - 4.87e56T + 1.03e115T^{2} \) |
| 97 | \( 1 + 9.59e57T + 1.65e117T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21383282055565486214759157458, −9.599210444913944875696031548827, −8.282043910103162341352320311914, −7.57836832179624098180211150388, −6.17952455407977586233642147469, −5.07743085524767562147638274710, −2.96871633952156879790967320802, −1.97387955578542324916137332478, −1.11246656105468285779664892044, 0,
1.11246656105468285779664892044, 1.97387955578542324916137332478, 2.96871633952156879790967320802, 5.07743085524767562147638274710, 6.17952455407977586233642147469, 7.57836832179624098180211150388, 8.282043910103162341352320311914, 9.599210444913944875696031548827, 10.21383282055565486214759157458
